This makes the rule \(\varrho ^{ }_{a}\colon \mathcal {A}\rightarrow \mathcal {A}^{+}\)  a continuous map, where the topology on the set \(\mathcal {A}^{+}\)  of finite words over \(\mathcal {A}\)  is the topology of a disjoint union; see [1]}. Moreover, the restrictions we impose on \({a}\)  imply that \(\varrho ^{ }_{a}\)  is a primitive substitution. In the compact alphabet setting, \(\varrho _{a}\)  being primitive means that, for any non-empty open set \(U\subset \mathcal {A}\) , there exists a power \(n:=n(U)\)  such that \(\varrho _{a}^{n}(b)\)  contains a letter in \(U\) , for all \(b\in \mathcal {A}\) ; see [2]}.
Note that such a rule results in a well-defined shift space \((X_{\varrho },S)\)  that has many nice properties of primitive substitutions on finite alphabets (e.g., minimality, unique ergodicity); see [2]}. For more details on dynamical properties of substitutions on compact alphabets, we refer the reader to [1]}.
